NO
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be disproven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 159 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) FilterProof [EQUIVALENT, 0 ms]
        (11) IntTRS
        (12) IntTRSPeriodicNontermProof [COMPLETE, 5 ms]
        (13) NO
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (20) NO


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(0)
Obligation:
Rules:
l0(nHAT0, xHAT0) -> l1(nHATpost, xHATpost) :|: xHAT0 = xHATpost && nHAT0 = nHATpost
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l2(x4, x5) -> l0(x6, x7) :|: x4 <= 0 && x8 = 0 && x7 = 1 && x4 = x6
l2(x9, x10) -> l3(x11, x12) :|: x10 = x12 && x9 = x11 && 1 <= x9
l3(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x13 = x15
l4(x17, x18) -> l2(x19, x20) :|: x20 = 1 && x19 = x19
l5(x21, x22) -> l4(x23, x24) :|: x22 = x24 && x21 = x23
Start term: l5(nHAT0, xHAT0)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
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(2)
Obligation:
Rules:
l0(nHAT0, xHAT0) -> l1(nHATpost, xHATpost) :|: xHAT0 = xHATpost && nHAT0 = nHATpost
l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
l2(x4, x5) -> l0(x6, x7) :|: x4 <= 0 && x8 = 0 && x7 = 1 && x4 = x6
l2(x9, x10) -> l3(x11, x12) :|: x10 = x12 && x9 = x11 && 1 <= x9
l3(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x13 = x15
l4(x17, x18) -> l2(x19, x20) :|: x20 = 1 && x19 = x19
l5(x21, x22) -> l4(x23, x24) :|: x22 = x24 && x21 = x23
Start term: l5(nHAT0, xHAT0)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) l0(nHAT0, xHAT0) -> l1(nHATpost, xHATpost) :|: xHAT0 = xHATpost && nHAT0 = nHATpost
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2
(3) l2(x4, x5) -> l0(x6, x7) :|: x4 <= 0 && x8 = 0 && x7 = 1 && x4 = x6
(4) l2(x9, x10) -> l3(x11, x12) :|: x10 = x12 && x9 = x11 && 1 <= x9
(5) l3(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x13 = x15
(6) l4(x17, x18) -> l2(x19, x20) :|: x20 = 1 && x19 = x19
(7) l5(x21, x22) -> l4(x23, x24) :|: x22 = x24 && x21 = x23

Arcs:
(1) -> (2)
(2) -> (1)
(3) -> (1)
(4) -> (5)
(5) -> (3), (4)
(6) -> (3), (4)
(7) -> (6)

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) l2(x9, x10) -> l3(x11, x12) :|: x10 = x12 && x9 = x11 && 1 <= x9
(2) l3(x13, x14) -> l2(x15, x16) :|: x14 = x16 && x13 = x15

Arcs:
(1) -> (2)
(2) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
l2(x11:0, x10:0) -> l2(x11:0, x10:0) :|: x11:0 > 0

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(8) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   l2(x1, x2) -> l2(x1)

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(9)
Obligation:
Rules:
l2(x11:0) -> l2(x11:0) :|: x11:0 > 0

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(10) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l2(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
l2(x11:0) -> l2(x11:0) :|: x11:0 > 0

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(12) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x11:0) -> f(1, x11:0) :|: pc = 1 && x11:0 > 0
Witness term starting non-terminating reduction: f(1, 1)
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(13)
NO

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(14)
Obligation:

Termination digraph:
Nodes:
(1) l0(nHAT0, xHAT0) -> l1(nHATpost, xHATpost) :|: xHAT0 = xHATpost && nHAT0 = nHATpost
(2) l1(x, x1) -> l0(x2, x3) :|: x1 = x3 && x = x2

Arcs:
(1) -> (2)
(2) -> (1)

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
l0(nHAT0:0, x3:0) -> l0(nHAT0:0, x3:0) :|: TRUE

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
l0(VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
l0(nHAT0:0, x3:0) -> l0(nHAT0:0, x3:0) :|: TRUE

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(19) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, nHAT0:0, x3:0) -> f(1, nHAT0:0, x3:0) :|: pc = 1 && TRUE
Witness term starting non-terminating reduction: f(1, -8, -8)
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(20)
NO